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A Square of Things Quadratic Equations

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“one square, and ten roots of the same, are equal to thirty-nine…what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?”

Algebra from the Beginning

Solutions in 825

No algebraic symbolism, thus all problems are like recipe cards

Solution: “you halve the number of the roots, which in the present instance yields five. This you multiple by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine.

Quadratic formula:

X= b 2 b

+ c -

2 2

Solutions Used Today

Early 17th Century mathematicians came up with algebraic symbols

Letters from the end = unknown numbers

Example: x, y, z

Letters from the beginning = known numbers

Example: a, b, c

Thomas Harriot and Rene Descartes rearranged equations so that they always equal 0.

Thus: ax2 + bx = c & ax2 + c = bx

Became ax2 + bx + c = 0

Solutions Today Cont.

Question: “one square, and ten roots of the same, are equal to thirty-nine…what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?

Translate:

Unknown: x “root of the square x2 “

“ten roots of the square”  10x

Equation: x2 + 10x = 39

Solution: “you halve the number of the roots, which in the present instance yields five. This you multiple by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three.”